| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2010 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Bivariate data |
| Type | Calculate r from raw bivariate data |
| Difficulty | Standard +0.3 This is a straightforward calculation of the product moment correlation coefficient (PMCC) from raw data using standard formulas. While it requires careful arithmetic with 12 data points and computing Sxx, Syy, and Sxy, it's a routine S1 procedure with no conceptual challenges. Part (d) is even easier as the summary statistics are provided. The interpretation aspects are basic. This is slightly easier than average due to being purely procedural. |
| Spec | 2.02c Scatter diagrams and regression lines2.02d Informal interpretation of correlation5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation |
| Item | Purchase price (£ \(\boldsymbol { x }\) ) | Auction price (£ y) |
| A | 20 | 30 |
| B | 35 | 45 |
| C | 18 | 25 |
| D | 50 | 50 |
| E | 45 | 38 |
| F | 55 | 45 |
| G | 43 | 50 |
| H | 81 | 90 |
| I | 90 | 85 |
| J | 30 | 190 |
| K | 57 | 65 |
| L | 112 | 25 |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt at substitution into correct corresponding formula for \(r\) | B3, (B2), (B1), (M1), (m1), (A1) | AWFW (\(-0.03546\)). AWFW. AWFW. 636 42702 738 68294 &38605 (all 5 attempted). 8994 22907 & \(-\)509 (all 3 attempted) |
| Answer | Marks | Guidance |
|---|---|---|
| between purchase and auction prices of antiques | B1dep, B1 | Dependent on \(-0.1 < r < 0.1\). Or equivalent; must qualify strength as 'zero'; B0dep for very weak/weak/etc unless then qualified correctly. Context; providing \(-1 < r < 1\) |
| Answer | Marks | Guidance |
|---|---|---|
| 3 correct labelled points | B3, (B2), (B1) | Deduct 1 mark if > 1 point not labelled or labelled incorrectly |
| Answer | Marks | Guidance |
|---|---|---|
| (Otherwise) a positive/linear correlation | B1, B1 | Or equivalent. Or equivalent; ignore any qualification of 'strength' |
| Answer | Marks | Guidance |
|---|---|---|
| \(r = 0.943 \text{ to } 0.944\) | M1, A1 | Used. Award B2 for a correct answer without/with different method. AWFW (0.94359) |
| Answer | Marks | Guidance |
|---|---|---|
| Previous calculation of \(r\) was not appropriate (due to outliers) | B1dep, (B1) | Dependent on \(0.9 < r < 1\). Or equivalent; must qualify strength and indicate positive; B0dep for high/etc. |
## Part (a)
$r = -0.0355 \text{ to } -0.035$
$r = -0.036 \text{ to } -0.034$
$r = -0.04$ to $+0.04$
or
Attempt at $\sum x, \sum x^2, \sum y, \sum y^2 \& \sum xy$
or
Attempt at $S_{xx}, S_{yy} \& S_{xy}$
or
Attempt at substitution into correct corresponding formula for $r$ | B3, (B2), (B1), (M1), (m1), (A1) | AWFW ($-0.03546$). AWFW. AWFW. 636 42702 738 68294 &38605 (all 5 attempted). 8994 22907 & $-$509 (all 3 attempted) | 3
## Part (b)
Almost/virtually/practically **no / zero** (linear) **correlation / relationship / association / link** (but not 'no trend')
between **purchase and auction prices** of antiques | B1dep, B1 | Dependent on $-0.1 < r < 0.1$. Or equivalent; must qualify strength as 'zero'; B0dep for very weak/weak/etc unless then qualified correctly. Context; providing $-1 < r < 1$ | 2
## Part (c)(i)
Figure 1:
6 correct labelled points
5 or 4 correct labelled points
3 correct labelled points | B3, (B2), (B1) | Deduct 1 mark if > 1 point not labelled or labelled incorrectly | 3
## Part (c)(ii)
(Two) outlier/anomaly/unusual or identification of J and L
(Otherwise) a positive/linear correlation | B1, B1 | Or equivalent. Or equivalent; ignore any qualification of 'strength' | 2
## Part (d)(i)
$r = \frac{4268.8}{\sqrt{4854.4 \times 4216.1}}$
$r = 0.943 \text{ to } 0.944$ | M1, A1 | Used. Award B2 for a correct answer without/with different method. AWFW (0.94359) | 2
## Part (d)(ii)
**Very strong/strong positive (linear) correlation/relationship/association/link**
Previous calculation of $r$ was not appropriate (due to outliers) | B1dep, (B1) | Dependent on $0.9 < r < 1$. Or equivalent; must qualify strength and indicate positive; B0dep for high/etc. | 1
---
**TOTAL: 75**7 [Figure 1, printed on the insert, is provided for use in this question.]\\
Harold considers himself to be an expert in assessing the auction value of antiques. He regularly visits car boot sales to buy items that he then sells at his local auction rooms.
Harold's father, Albert, who is not convinced of his son's expertise, collects the following data from a random sample of 12 items bought by Harold.
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
Item & Purchase price (£ $\boldsymbol { x }$ ) & Auction price (£ y) \\
\hline
A & 20 & 30 \\
\hline
B & 35 & 45 \\
\hline
C & 18 & 25 \\
\hline
D & 50 & 50 \\
\hline
E & 45 & 38 \\
\hline
F & 55 & 45 \\
\hline
G & 43 & 50 \\
\hline
H & 81 & 90 \\
\hline
I & 90 & 85 \\
\hline
J & 30 & 190 \\
\hline
K & 57 & 65 \\
\hline
L & 112 & 25 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Calculate the value of the product moment correlation coefficient between $x$ and $y$.
\item Interpret your value in the context of this question.
\item \begin{enumerate}[label=(\roman*)]
\item On Figure 1, complete the scatter diagram for these data.
\item Comment on what this reveals.
\end{enumerate}\item When items J and L are omitted from the data, it is found that
$$S _ { x x } = 4854.4 \quad S _ { y y } = 4216.1 \quad S _ { x y } = 4268.8$$
\begin{enumerate}[label=(\roman*)]
\item Calculate the value of the product moment correlation coefficient between $x$ and $y$ for the remaining 10 items.
\item Hence revise as necessary your interpretation in part (b).
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S1 2010 Q7 [13]}}